Finite Group Databasebeta

← #27466302

Subgroups — #27466302

Order 13568.

872 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278417C211yesmaximal, characteristic, complemented
2#20276272C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20276118C211yesmaximal, characteristic, complemented
5#20275851C211yesmaximal, characteristic, complemented
6#20275675C211yesmaximal, characteristic, complemented
7#20275401C211yesmaximal, characteristic, complemented
8#20275401C211yesmaximal, characteristic, complemented
9#20275144C211yesmaximal, characteristic
10#20275033C211yesmaximal, characteristic
11#20274842C211yesmaximal, characteristic, complemented
12#20274801C211yesmaximal, characteristic, complemented
13C22⋊C4×#88322C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#2501C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133194—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810555—21no—
28#16810544—21no—
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810498—21no—
32#16810493C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#123×C53—21no—
49C2×#123×C53—21no—
50C2×#121×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#399—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#384C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133217—22no—
69#13133217—41no—
70#13133194—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84#13133138—22no—
85#13133138—22no—
86#13133138—41no—
87#13133138—41no—
88C22⋊#88322—22no—
89C4×#88322—22no—
90C4×#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×D4×C53—21no—
95C22×D4×C53C2311yescharacteristic
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98#123×C53—44no—
99#123×C53—44no—
100#121×C53—22no—
101#121×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53D411yescharacteristic
122#95×C53D411yescharacteristic
123C24×C53—21no—
124C24×C53C2×D411yescharacteristic
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C2×D4×C53—42no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53#4112yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C2×Dic106—42no—
160C2×Dic106—42no—
161C2×Dic106—81no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53D814yes—
179C22⋊C4×C53SD1614yes—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C22⋊Dic53—42no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196C4⋊Dic53#4112yes—
197#11719919—42no—
198#11719919—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—44no—
211C2×#88322#4112yes—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53#12412yes—
224C23×C53C2×D812yes—
225C23×C53C2×SD1612yes—
226C23×C53C22×D411yescharacteristic
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53—84no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—84no—
256C2×C4×C53#12412yes—
257C2×C4×C53C2×D812yes—
258C2×C4×C53C2×D812yes—
259C2×C4×C53C2×SD1612yes—
260C2×C4×C53C2×SD1612yes—
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53H(5,𝔽2)11yescharacteristic
266C2×C4×C532−1+411yescharacteristic
267C2×Dic53—22no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—84no—
280Dic106—84no—
281#88322—22no—
282#88322—41no—
283#88322—41no—
284#88322—41no—
285#27695—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53#48614yes—
300C22×C53C2×#12411yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#54711yescharacteristic
303C22×C53C22×D811yescharacteristic
304C22×C53C22×SD1611yescharacteristic
305C22×C53#57711yescharacteristic
306C22×C53#57811yescharacteristic
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53#52911yescharacteristic
316C4×C53#53911yescharacteristic
317C4×C53#55111yescharacteristic
318C4×C53#55211yescharacteristic
319Dic53—42no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323#2501D5311yescharacteristic
324#2440—531no—
325#2099—531no—
326#2099—531no—
327#2071—531no—
328#2023—531no—
329#1990—531no—
330#1934—531no—
331#1934—531no—
332#1893—531no—
333#1874—531no—
334#1834—531no—
335#1823—531no—
336C8×C22⋊C4—531no—
337C2×#340—531no—
338C2×C53—24no—
339C2×C53—161no—
340C2×C53#248811yescharacteristic
341C2×C53C2×#48611yescharacteristic
342C2×C53#313711yescharacteristic
343C2×C53#329411yescharacteristic
344C2×C53#329811yescharacteristic
345C2×C53#333411yescharacteristic
346C2×C53#333511yescharacteristic
347C23×C8—531no—
348C2×C4⋊Q8—531no—
349C2×#123—21no—
350C2×#123—21no—
351C2×C22⋊Q8—1061no—
352C2×#121C2×D5311yescharacteristic
353C22×C4⋊C4—531no—
354C22×C4⋊C4C2×D5311yescharacteristic
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×C22⋊C8—531no—
360C2×C4×C8—531no—
361C2×C4×C8—531no—
362#399—21no—
363#399—21no—
364#398—1061no—
365#395—1061no—
366#390C2×D5311yescharacteristic
367#390C2×D5311yescharacteristic
368#389—531no—
369#389—531no—
370#385—1061no—
371#384—531no—
372#384C2×D5311yescharacteristic
373#384C2×D5311yescharacteristic
374#380—531no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336—531no—
389#336—531no—
390C53#2769511yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—21no—
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—1062no—
402C22×C8—1062no—
403C22×C8—2121no—
404C4⋊Q8—1062no—
405C4⋊Q8—2121no—
406#123—44no—
407#123—44no—
408C22⋊Q8—2124no—
409#121—22no—
410#121—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—2121no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C22⋊C4C22⋊D5311yescharacteristic
441C2×C42C22×D5311yescharacteristic
442C2×C42C22×D5311yescharacteristic
443#102—1062no—
444#102—1062no—
445#102—2121no—
446#102—2121no—
447C22⋊C8—1062no—
448C4×C8—1062no—
449C4×C8—2121no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95C22⋊D5311yescharacteristic
454#95C22⋊D5311yescharacteristic
455C24—21no—
456C24C2×C22⋊D5311yescharacteristic
457C2×Q8—4241no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—4241no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4#1171994712yes—
487C22×C4#1171994712yes—
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494C2×C8—532no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—2124no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—532no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—4241no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—2122no—
538C22⋊C4#1171992214yes—
539C22⋊C4#1171992314yes—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#1313320412yes—
556C23C2×#1171992212yes—
557C23C2×#1171992312yes—
558C23C22×C22⋊D5311yescharacteristic
559Q8—4244no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—1062no—
590C2×C4—4244no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4#1313320412yes—
603C2×C4C2×#1171992212yes—
604C2×C4C2×#1171992212yes—
605C2×C4C2×#1171992312yes—
606C2×C4C2×#1171992312yes—
607C2×C4#41×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326311yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—1062no—
614C8—2121no—
615C8—2121no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22#1681096914yes—
631C22C2×#1313320411yescharacteristic
632C22#1681126111yescharacteristic
633C22#1681127211yescharacteristic
634C22C22×#1171992211yescharacteristic
635C22C22×#1171992311yescharacteristic
636C22#1681143111yescharacteristic
637C22#1681143811yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—4241no—
647C4—4241no—
648C4—4241no—
649C4—2122no—
650C4#1681115011yescharacteristic
651C4#1681121711yescharacteristic
652C4#1681129611yescharacteristic
653C4#1681131011yescharacteristic
654C2—24no—
655C2—161no—
656C2#2027904511yescharacteristic
657C2C2×#1681096911yescharacteristic
658C2#2028508011yescharacteristic
659C2#2028652211yescharacteristic
660C2#2028657411yescharacteristic
661C2#2028705811yescharacteristic
662C2#2028707211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2501C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2501D53
hypocenterC53#27695
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2501

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20278417C2semidirect
2. #20276272C2semidirect
3. #20276272C2semidirect
4. #20276118C2semidirect
5. #20275851C2semidirect
6. #20275675C2semidirect
7. #20275401C2semidirect
8. #20275401C2semidirect
11. #20274842C2semidirect
12. #20274801C2semidirect
13. C22⋊C4×#88322C2semidirect
14. C2×#16810385C2semidirect
390. C53#27695semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202784172
#202762722
#202761182
#202758512
#202756752
#202754012
#202751442
#202750332
#202748422
#202748012
C22⋊C4×#883222
C2×#168103852
C53×#25012
#2769553